| L(s) = 1 | + (−1.39 − 0.213i)2-s + (−1.12 − 0.467i)3-s + (1.90 + 0.597i)4-s + (−0.906 + 0.462i)5-s + (1.47 + 0.894i)6-s + (1.47 + 2.41i)7-s + (−2.54 − 1.24i)8-s + (−1.06 − 1.06i)9-s + (1.36 − 0.452i)10-s + (5.44 + 0.428i)11-s + (−1.87 − 1.56i)12-s + (1.37 + 5.73i)13-s + (−1.54 − 3.68i)14-s + (1.23 − 0.0975i)15-s + (3.28 + 2.28i)16-s + (5.06 − 4.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.151i)2-s + (−0.651 − 0.269i)3-s + (0.954 + 0.298i)4-s + (−0.405 + 0.206i)5-s + (0.603 + 0.365i)6-s + (0.558 + 0.910i)7-s + (−0.898 − 0.439i)8-s + (−0.355 − 0.355i)9-s + (0.432 − 0.143i)10-s + (1.64 + 0.129i)11-s + (−0.541 − 0.452i)12-s + (0.381 + 1.59i)13-s + (−0.414 − 0.984i)14-s + (0.320 − 0.0251i)15-s + (0.821 + 0.570i)16-s + (1.22 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620926 + 0.128574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620926 + 0.128574i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.213i)T \) |
| 41 | \( 1 + (6.40 - 0.00937i)T \) |
| good | 3 | \( 1 + (1.12 + 0.467i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.906 - 0.462i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-1.47 - 2.41i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-5.44 - 0.428i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 5.73i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-5.06 + 4.32i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.80 - 0.912i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (4.52 - 3.29i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.13 - 0.968i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (2.34 + 7.21i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.577 + 1.77i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.547 - 0.0867i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (1.86 - 3.04i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (4.02 - 4.71i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-0.796 - 1.09i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.44 + 1.33i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.660 + 8.39i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (0.307 - 3.91i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-2.33 + 2.33i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.36 + 5.70i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 1.36iT - 83T^{2} \) |
| 89 | \( 1 + (-12.4 + 7.61i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.242 + 3.08i)T + (-95.8 + 15.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.04193032842503955712830910269, −11.72582795742417388304155629353, −11.42243867247193914420750190200, −9.537433214975137107974715505517, −9.096308394718255817330002135402, −7.72498108518056543387248563784, −6.68307627371036219577062501280, −5.69620637411749891494815005220, −3.61208148187392378718202852706, −1.56938002697982126689898392741,
1.05805826579338069117661790059, 3.66280081936950359338143371337, 5.37575893553515408982457767673, 6.47668476499186490941863726988, 7.88420425416975905938637759654, 8.418577661464632448078297746626, 10.03740166187545157241832888968, 10.59757622983985866635452053218, 11.58815782522071765343209579302, 12.27310778709064734867500233819
